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.\" ========================================================================
.\"
.IX Title "INTEG 3pm"
.TH INTEG 3pm 2024-05-12 "perl v5.38.2" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
.\" way too many mistakes in technical documents.
.if n .ad l
.nh
.SH NAME
PDL::GSL::INTEG \- PDL interface to numerical integration routines in GSL
.SH DESCRIPTION
.IX Header "DESCRIPTION"
This is an interface to the numerical integration package present in the
GNU Scientific Library, which is an implementation of QUADPACK.
.PP
Functions are named \fBgslinteg_{algorithm}\fR where {algorithm}
is the QUADPACK naming convention. The available functions are:
.IP "gslinteg_qng: Non-adaptive Gauss-Kronrod integration" 3
.IX Item "gslinteg_qng: Non-adaptive Gauss-Kronrod integration"
.PD 0
.IP "gslinteg_qag: Adaptive integration" 3
.IX Item "gslinteg_qag: Adaptive integration"
.IP "gslinteg_qags: Adaptive integration with singularities" 3
.IX Item "gslinteg_qags: Adaptive integration with singularities"
.IP "gslinteg_qagp: Adaptive integration with known singular points" 3
.IX Item "gslinteg_qagp: Adaptive integration with known singular points"
.IP "gslinteg_qagi: Adaptive integration on infinite interval of the form (\-\einfty,\einfty)" 3
.IX Item "gslinteg_qagi: Adaptive integration on infinite interval of the form (-infty,infty)"
.IP "gslinteg_qagiu: Adaptive integration on infinite interval of the form (la,\einfty)" 3
.IX Item "gslinteg_qagiu: Adaptive integration on infinite interval of the form (la,infty)"
.IP "gslinteg_qagil: Adaptive integration on infinite interval of the form (\-\einfty,lb)" 3
.IX Item "gslinteg_qagil: Adaptive integration on infinite interval of the form (-infty,lb)"
.IP "gslinteg_qawc: Adaptive integration for Cauchy principal values" 3
.IX Item "gslinteg_qawc: Adaptive integration for Cauchy principal values"
.IP "gslinteg_qaws: Adaptive integration for singular functions" 3
.IX Item "gslinteg_qaws: Adaptive integration for singular functions"
.IP "gslinteg_qawo: Adaptive integration for oscillatory functions" 3
.IX Item "gslinteg_qawo: Adaptive integration for oscillatory functions"
.IP "gslinteg_qawf: Adaptive integration for Fourier integrals" 3
.IX Item "gslinteg_qawf: Adaptive integration for Fourier integrals"
.PD
.PP
Each algorithm computes an approximation to the integral, I,
of the function f(x)w(x), where w(x) is a weight function
(for general integrands w(x)=1). The user provides absolute
and relative error bounds (epsabs,epsrel) which specify
the following accuracy requirement:
.PP
|RESULT \- I|  <= max(epsabs, epsrel |I|)
.PP
The routines will fail to converge if the
error bounds are too stringent, but always return the best
approximation obtained up to that stage
.PP
All functions return the result, and estimate of the absolute
error and an error flag (which is zero if there were no problems).
You are responsible for checking for any errors, no warnings are issued
unless the option {Warn => 'y'} is specified in which case
the reason of failure will be printed.
.PP
You can nest integrals up to 20 levels. If you find yourself in
the unlikely situation that you need more, you can change the value
of 'max_nested_integrals' in the first line of the file 'FUNC.c'
and recompile.
.SH NOMENCLATURE
.IX Header "NOMENCLATURE"
Throughout this documentation we strive to use the same variables that
are present in the original GSL documentation (see See
Also). Oftentimes those variables are called \f(CW\*(C`a\*(C'\fR and
\&\f(CW\*(C`b\*(C'\fR. Since good Perl coding practices discourage the use of Perl
variables \f(CW$a\fR and \f(CW$b\fR, here we refer to Parameters \f(CW\*(C`a\*(C'\fR and \f(CW\*(C`b\*(C'\fR
as \f(CW$pa\fR and \f(CW$pb\fR, respectively, and Limits (of domain or
integration) as \f(CW$la\fR and \f(CW$lb\fR.
.SH SYNOPSIS
.IX Header "SYNOPSIS"
.Vb 2
\&   use PDL;
\&   use PDL::GSL::INTEG;
\&
\&   my $la = 1.2;
\&   my $lb = 3.7;
\&   my $epsrel = 0;
\&   my $epsabs = 1e\-6;
\&
\&   # Non adaptive integration
\&   my ($res,$abserr,$ierr,$neval) = gslinteg_qng(\e&myf,$la,$lb,$epsrel,$epsabs);
\&   # Warnings on
\&   my ($res,$abserr,$ierr,$neval) = gslinteg_qng(\e&myf,$la,$lb,$epsrel,$epsabs,{Warn=>\*(Aqy\*(Aq});
\&
\&   # Adaptive integration with warnings on
\&   my $limit = 1000;
\&   my $key = 5;
\&   my ($res,$abserr,$ierr) = gslinteg_qag(\e&myf,$la,$lb,$epsrel,
\&                                     $epsabs,$limit,$key,{Warn=>\*(Aqy\*(Aq});
\&
\&   sub myf{
\&     my ($x) = @_;
\&     return exp(\-$x**2);
\&   }
\&#line 127 "INTEG.pm"
.Ve
.SH FUNCTIONS
.IX Header "FUNCTIONS"
.SS gslinteg_qng
.IX Subsection "gslinteg_qng"
.Vb 2
\&  Signature: (a(); b(); epsabs(); epsrel(); int gslwarn();
\&                   [o] result(); [o] abserr(); int [o] neval(); int [o] ierr();; SV* function)
.Ve
.PP
Non-adaptive Gauss-Kronrod integration
.PP
This function applies the Gauss-Kronrod 10\-point, 21\-point, 43\-point
and 87\-point integration rules in succession until an estimate of the
integral of f over ($la,$lb) is achieved within the desired absolute
and relative error limits, \f(CW$epsabs\fR and \f(CW$epsrel\fR.  It is meant for fast
integration of smooth functions. It returns an array with the result,
an estimate of the absolute error, an error flag and the number of
function evaluations performed.
.PP
Usage:
.PP
.Vb 2
\&  ($res,$abserr,$ierr,$neval) = gslinteg_qng($function_ref,$la,$lb,
\&                                             $epsrel,$epsabs,[{Warn => $warn}]);
.Ve
.PP
Example:
.PP
.Vb 3
\&   my ($res,$abserr,$ierr,$neval) = gslinteg_qng(\e&f,0,1,0,1e\-9);
\&   # with warnings on
\&   my ($res,$abserr,$ierr,$neval) = gslinteg_qng(\e&f,0,1,0,1e\-9,{Warn => \*(Aqy\*(Aq});
\&
\&   sub f{
\&     my ($x) = @_;
\&     return ($x**2.6)*log(1.0/$x);
\&   }
.Ve
.PP
gslinteg_qng does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS gslinteg_qag
.IX Subsection "gslinteg_qag"
.Vb 3
\&  Signature: (a(); b(); epsabs();epsrel();
\&                   int limit(); int key(); int n(); int gslwarn();
\&                   [o] result(); [o] abserr(); int [o] ierr();; SV* function)
.Ve
.PP
Adaptive integration
.PP
This function applies an integration rule adaptively until an estimate of
the integral of f over ($la,$lb) is achieved within the desired absolute and
relative error limits, \f(CW$epsabs\fR and \f(CW$epsrel\fR. On each iteration the adaptive
integration strategy bisects the interval with the largest error estimate;
the maximum number of allowed subdivisions is given by the parameter \f(CW$limit\fR.
The integration rule is determined by the
value of \f(CW$key\fR, which has to be one of (1,2,3,4,5,6) and correspond to
the 15, 21, 31, 41, 51 and 61  point Gauss-Kronrod rules respectively.
It returns an array with the result, an estimate of the absolute error
and an error flag.
.PP
Usage:
.PP
.Vb 2
\&  ($res,$abserr,$ierr) = gslinteg_qag($function_ref,$la,$lb,$epsrel,
\&                                      $epsabs,$limit,$key,[{Warn => $warn}]);
.Ve
.PP
Example:
.PP
.Vb 3
\&  my ($res,$abserr,$ierr) = gslinteg_qag(\e&f,0,1,0,1e\-10,1000,1);
\&  # with warnings on
\&  my ($res,$abserr,$ierr) = gslinteg_qag(\e&f,0,1,0,1e\-10,1000,1,{Warn => \*(Aqy\*(Aq});
\&
\&  sub f{
\&     my ($x) = @_;
\&     return ($x**2.6)*log(1.0/$x);
\&   }
.Ve
.PP
gslinteg_qag does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS gslinteg_qags
.IX Subsection "gslinteg_qags"
.Vb 3
\&  Signature: (a(); b(); epsabs(); epsrel();
\&                   int limit(); int n(); int gslwarn();
\&                   [o] result(); [o] abserr(); int [o] ierr();; SV* function)
.Ve
.PP
Adaptive integration with singularities
.PP
This function applies the Gauss-Kronrod 21\-point integration rule
adaptively until an estimate of the integral of f over ($la,$lb) is
achieved within the desired absolute and relative error limits,
\&\f(CW$epsabs\fR and \f(CW$epsrel\fR. The algorithm is such that it
accelerates the convergence of the integral in the presence of
discontinuities and integrable singularities.
The maximum number of allowed subdivisions done by the adaptive
algorithm must be supplied in the parameter \f(CW$limit\fR.
.PP
Usage:
.PP
.Vb 2
\&  ($res,$abserr,$ierr) = gslinteg_qags($function_ref,$la,$lb,$epsrel,
\&                                       $epsabs,$limit,[{Warn => $warn}]);
.Ve
.PP
Example:
.PP
.Vb 3
\&  my ($res,$abserr,$ierr) = gslinteg_qags(\e&f,0,1,0,1e\-10,1000);
\&  # with warnings on
\&  ($res,$abserr,$ierr) = gslinteg_qags(\e&f,0,1,0,1e\-10,1000,{Warn => \*(Aqy\*(Aq});
\&
\&  sub f{
\&     my ($x) = @_;
\&     return ($x)*log(1.0/$x);
\&   }
.Ve
.PP
gslinteg_qags does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS gslinteg_qagp
.IX Subsection "gslinteg_qagp"
.Vb 2
\&  Signature: (pts(l); epsabs(); epsrel();int limit();int n();int gslwarn();
\&                   [o] result(); [o] abserr();int [o] ierr();; SV* function)
.Ve
.PP
Adaptive integration with known singular points
.PP
This function applies the adaptive integration algorithm used by
gslinteg_qags taking into account the location of singular points
until an estimate of
the integral of f over ($la,$lb) is achieved within the desired absolute and
relative error limits, \f(CW$epsabs\fR and \f(CW$epsrel\fR.
Singular points are supplied in the ndarray \f(CW$points\fR, whose endpoints
determine the integration range.
So, for example, if the function has singular points at x_1 and x_2 and the
integral is desired from a to b (a < x_1 < x_2 < b), \f(CW$points\fR = pdl(a,x_1,x_2,b).
The maximum number of allowed subdivisions done by the adaptive
algorithm must be supplied in the parameter \f(CW$limit\fR.
.PP
Usage:
.PP
.Vb 2
\&  ($res,$abserr,$ierr) = gslinteg_qagp($function_ref,$points,$epsabs,
\&                                       $epsrel,$limit,[{Warn => $warn}])
.Ve
.PP
Example:
.PP
.Vb 4
\&  my $points = pdl(0,1,sqrt(2),3);
\&  my ($res,$abserr,$ierr) = gslinteg_qagp(\e&f,$points,0,1e\-3,1000);
\&  # with warnings on
\&  ($res,$abserr,$ierr) = gslinteg_qagp(\e&f,$points,0,1e\-3,1000,{Warn => \*(Aqy\*(Aq});
\&
\&  sub f{
\&    my ($x) = @_;
\&    my $x2 = $x**2;
\&    my $x3 = $x**3;
\&    return $x3 * log(abs(($x2\-1.0)*($x2\-2.0)));
\&  }
.Ve
.PP
gslinteg_qagp does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS gslinteg_qagi
.IX Subsection "gslinteg_qagi"
.Vb 2
\&  Signature: (epsabs(); epsrel(); int limit(); int n();int gslwarn();
\&                   [o] result(); [o] abserr(); int [o] ierr();; SV* function)
.Ve
.PP
Adaptive integration on infinite interval
.PP
This function estimates the integral of the function f over the
infinite interval (\-\einfty,+\einfty) within the desired absolute and
relative error limits, \f(CW$epsabs\fR and \f(CW$epsrel\fR.
After a transformation, the algorithm
of gslinteg_qags with a 15\-point Gauss-Kronrod rule is used.
The maximum number of allowed subdivisions done by the adaptive
algorithm must be supplied in the parameter \f(CW$limit\fR.
.PP
Usage:
.PP
.Vb 2
\&  ($res,$abserr,$ierr) = gslinteg_qagi($function_ref,$epsabs,
\&                                       $epsrel,$limit,[{Warn => $warn}]);
.Ve
.PP
Example:
.PP
.Vb 3
\&  my ($res,$abserr,$ierr) = gslinteg_qagi(\e&myfn,1e\-7,0,1000);
\&  # with warnings on
\&  ($res,$abserr,$ierr) = gslinteg_qagi(\e&myfn,1e\-7,0,1000,{Warn => \*(Aqy\*(Aq});
\&
\&  sub myfn{
\&    my ($x) = @_;
\&    return exp(\-$x \- $x*$x) ;
\&  }
.Ve
.PP
gslinteg_qagi does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS gslinteg_qagiu
.IX Subsection "gslinteg_qagiu"
.Vb 2
\&  Signature: (a(); epsabs(); epsrel();int limit();int n();int gslwarn();
\&                   [o] result(); [o] abserr();int [o] ierr();; SV* function)
.Ve
.PP
Adaptive integration on infinite interval
.PP
This function estimates the integral of the function f over the
infinite interval (la,+\einfty) within the desired absolute and
relative error limits, \f(CW$epsabs\fR and \f(CW$epsrel\fR.
After a transformation, the algorithm
of gslinteg_qags with a 15\-point Gauss-Kronrod rule is used.
The maximum number of allowed subdivisions done by the adaptive
algorithm must be supplied in the parameter \f(CW$limit\fR.
.PP
Usage:
.PP
.Vb 2
\&  ($res,$abserr,$ierr) = gslinteg_qagiu($function_ref,$la,$epsabs,
\&                                        $epsrel,$limit,[{Warn => $warn}]);
.Ve
.PP
Example:
.PP
.Vb 4
\&  my $alfa = 1;
\&  my ($res,$abserr,$ierr) = gslinteg_qagiu(\e&f,99.9,1e\-7,0,1000);
\&  # with warnings on
\&  ($res,$abserr,$ierr) = gslinteg_qagiu(\e&f,99.9,1e\-7,0,1000,{Warn => \*(Aqy\*(Aq});
\&
\&  sub f{
\&    my ($x) = @_;
\&    if (($x==0) && ($alfa == 1)) {return 1;}
\&    if (($x==0) && ($alfa > 1)) {return 0;}
\&    return ($x**($alfa\-1))/((1+10*$x)**2);
\&  }
.Ve
.PP
gslinteg_qagiu does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS gslinteg_qagil
.IX Subsection "gslinteg_qagil"
.Vb 2
\&  Signature: (b(); epsabs(); epsrel();int limit();int n();int gslwarn();
\&                   [o] result(); [o] abserr();int [o] ierr();; SV* function)
.Ve
.PP
Adaptive integration on infinite interval
.PP
This function estimates the integral of the function f over the
infinite interval (\-\einfty,lb) within the desired absolute and
relative error limits, \f(CW$epsabs\fR and \f(CW$epsrel\fR.
After a transformation, the algorithm
of gslinteg_qags with a 15\-point Gauss-Kronrod rule is used.
The maximum number of allowed subdivisions done by the adaptive
algorithm must be supplied in the parameter \f(CW$limit\fR.
.PP
Usage:
.PP
.Vb 2
\&  ($res,$abserr,$ierr) = gslinteg_qagl($function_ref,$lb,$epsabs,
\&                                       $epsrel,$limit,[{Warn => $warn}]);
.Ve
.PP
Example:
.PP
.Vb 3
\&  my ($res,$abserr,$ierr) = gslinteg_qagil(\e&myfn,1.0,1e\-7,0,1000);
\&  # with warnings on
\&  ($res,$abserr,$ierr) = gslinteg_qagil(\e&myfn,1.0,1e\-7,0,1000,{Warn => \*(Aqy\*(Aq});
\&
\&  sub myfn{
\&    my ($x) = @_;
\&    return exp($x);
\&  }
.Ve
.PP
gslinteg_qagil does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS gslinteg_qawc
.IX Subsection "gslinteg_qawc"
.Vb 2
\&  Signature: (a(); b(); c(); epsabs(); epsrel();int limit();int n();int gslwarn();
\&                   [o] result(); [o] abserr();int [o] ierr();; SV* function)
.Ve
.PP
Adaptive integration for Cauchy principal values
.PP
This function computes the Cauchy principal value of the integral of f over (la,lb),
with a singularity at c, I = \eint_{la}^{lb} dx f(x)/(x \- c). The integral is
estimated within the desired absolute and relative error limits, \f(CW$epsabs\fR and \f(CW$epsrel\fR.
The maximum number of allowed subdivisions done by the adaptive
algorithm must be supplied in the parameter \f(CW$limit\fR.
.PP
Usage:
.PP
.Vb 1
\&  ($res,$abserr,$ierr) = gslinteg_qawc($function_ref,$la,$lb,$c,$epsabs,$epsrel,$limit)
.Ve
.PP
Example:
.PP
.Vb 3
\&  my ($res,$abserr,$ierr) = gslinteg_qawc(\e&f,\-1,5,0,0,1e\-3,1000);
\&  # with warnings on
\&  ($res,$abserr,$ierr) = gslinteg_qawc(\e&f,\-1,5,0,0,1e\-3,1000,{Warn => \*(Aqy\*(Aq});
\&
\&  sub f{
\&    my ($x) = @_;
\&    return 1.0 / (5.0 * $x * $x * $x + 6.0) ;
\&  }
.Ve
.PP
gslinteg_qawc does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS gslinteg_qaws
.IX Subsection "gslinteg_qaws"
.Vb 3
\&  Signature: (a(); b(); epsabs(); epsrel();int limit();
\&                 int n(); alpha(); beta(); int mu(); int nu(); int gslwarn();
\&                 [o] result(); [o] abserr(); int [o] ierr();; SV* function)
.Ve
.PP
Adaptive integration for singular functions
.PP
The algorithm in gslinteg_qaws is designed for integrands with algebraic-logarithmic
singularities at the end-points of an integration region.
Specifically, this function computes the integral given by
I = \eint_{la}^{lb} dx f(x) (x\-la)^alpha (lb\-x)^beta log^mu (x\-la) log^nu (lb-x).
The integral is
estimated within the desired absolute and relative error limits, \f(CW$epsabs\fR and \f(CW$epsrel\fR.
The maximum number of allowed subdivisions done by the adaptive
algorithm must be supplied in the parameter \f(CW$limit\fR.
.PP
Usage:
.PP
.Vb 3
\&  ($res,$abserr,$ierr) =
\&      gslinteg_qaws($function_ref,$alpha,$beta,$mu,$nu,$la,$lb,
\&                    $epsabs,$epsrel,$limit,[{Warn => $warn}]);
.Ve
.PP
Example:
.PP
.Vb 3
\&  my ($res,$abserr,$ierr) = gslinteg_qaws(\e&f,0,0,1,0,0,1,0,1e\-7,1000);
\&  # with warnings on
\&  ($res,$abserr,$ierr) = gslinteg_qaws(\e&f,0,0,1,0,0,1,0,1e\-7,1000,{Warn => \*(Aqy\*(Aq});
\&
\&  sub f{
\&    my ($x) = @_;
\&    if($x==0){return 0;}
\&    else{
\&      my $u = log($x);
\&      my $v = 1 + $u*$u;
\&      return 1.0/($v*$v);
\&    }
\&  }
.Ve
.PP
gslinteg_qaws does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS gslinteg_qawo
.IX Subsection "gslinteg_qawo"
.Vb 3
\&  Signature: (a(); b(); epsabs(); epsrel();int limit();int n();
\&                 int sincosopt(); omega(); L(); int nlevels();int gslwarn();
\&                 [o] result(); [o] abserr();int [o] ierr();; SV* function)
.Ve
.PP
Adaptive integration for oscillatory functions
.PP
This function uses an adaptive algorithm to compute the integral of f over
(la,lb) with the weight function sin(omega*x) or cos(omega*x) \-\- which of
sine or cosine is used is determined by the parameter \f(CW$opt\fR ('cos' or 'sin').
The integral is
estimated within the desired absolute and relative error limits, \f(CW$epsabs\fR and \f(CW$epsrel\fR.
The maximum number of allowed subdivisions done by the adaptive
algorithm must be supplied in the parameter \f(CW$limit\fR.
.PP
Usage:
.PP
.Vb 2
\&  ($res,$abserr,$ierr) = gslinteg_qawo($function_ref,$omega,$sin_or_cos,
\&                                $la,$lb,$epsabs,$epsrel,$limit,[opt])
.Ve
.PP
Example:
.PP
.Vb 4
\&  my $PI = 3.14159265358979323846264338328;
\&  my ($res,$abserr,$ierr) = PDL::GSL::INTEG::gslinteg_qawo(\e&f,10*$PI,\*(Aqsin\*(Aq,0,1,0,1e\-7,1000);
\&  # with warnings on
\&  ($res,$abserr,$ierr) = PDL::GSL::INTEG::gslinteg_qawo(\e&f,10*$PI,\*(Aqsin\*(Aq,0,1,0,1e\-7,1000,{Warn => \*(Aqy\*(Aq});
\&
\&  sub f{
\&    my ($x) = @_;
\&    if($x==0){return 0;}
\&    else{ return log($x);}
\&  }
.Ve
.PP
gslinteg_qawo does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS gslinteg_qawf
.IX Subsection "gslinteg_qawf"
.Vb 3
\&  Signature: (a(); epsabs();int limit();int n();
\&                 int sincosopt(); omega(); int nlevels();int gslwarn();
\&                 [o] result(); [o] abserr();int [o] ierr();; SV* function)
.Ve
.PP
Adaptive integration for Fourier integrals
.PP
This function attempts to compute a Fourier integral of the function
f over the semi-infinite interval [la,+\einfty). Specifically, it attempts
tp compute I = \eint_{la}^{+\einfty} dx f(x)w(x), where w(x) is sin(omega*x)
or cos(omega*x) \-\- which of sine or cosine is used is determined by
the parameter \f(CW$opt\fR ('cos' or 'sin').
The integral is
estimated within the desired absolute error limit \f(CW$epsabs\fR.
The maximum number of allowed subdivisions done by the adaptive
algorithm must be supplied in the parameter \f(CW$limit\fR.
.PP
Usage:
.PP
.Vb 1
\&  gslinteg_qawf($function_ref,$omega,$sin_or_cos,$la,$epsabs,$limit,[opt])
.Ve
.PP
Example:
.PP
.Vb 3
\&  my ($res,$abserr,$ierr) = gslinteg_qawf(\e&f,$PI/2.0,\*(Aqcos\*(Aq,0,1e\-7,1000);
\&  # with warnings on
\&  ($res,$abserr,$ierr) = gslinteg_qawf(\e&f,$PI/2.0,\*(Aqcos\*(Aq,0,1e\-7,1000,{Warn => \*(Aqy\*(Aq});
\&
\&  sub f{
\&    my ($x) = @_;
\&    if ($x == 0){return 0;}
\&    return 1.0/sqrt($x)
\&  }
.Ve
.PP
gslinteg_qawf does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SH BUGS
.IX Header "BUGS"
Feedback is welcome. Log bugs in the PDL bug database (the
database is always linked from <http://pdl.perl.org>).
.SH "SEE ALSO"
.IX Header "SEE ALSO"
PDL
.PP
The GSL documentation for numerical integration is online at
<https://www.gnu.org/software/gsl/doc/html/integration.html>
.SH AUTHOR
.IX Header "AUTHOR"
This file copyright (C) 2003,2005 Andres Jordan <ajordan@eso.org>
All rights reserved. There is no warranty. You are allowed to redistribute
this software documentation under certain conditions. For details, see the file
COPYING in the PDL distribution. If this file is separated from the
PDL distribution, the copyright notice should be included in the file.
.PP
The GSL integration routines were written by Brian Gough. QUADPACK
was written by Piessens, Doncker-Kapenga, Uberhuber and Kahaner.